Step | Hyp | Ref
| Expression |
1 | | frgpup3.k |
. . 3
⊢ (𝜑 → 𝐾 ∈ (𝐺 GrpHom 𝐻)) |
2 | | frgpup.x |
. . . 4
⊢ 𝑋 = (Base‘𝐺) |
3 | | frgpup.b |
. . . 4
⊢ 𝐵 = (Base‘𝐻) |
4 | 2, 3 | ghmf 18826 |
. . 3
⊢ (𝐾 ∈ (𝐺 GrpHom 𝐻) → 𝐾:𝑋⟶𝐵) |
5 | | ffn 6593 |
. . 3
⊢ (𝐾:𝑋⟶𝐵 → 𝐾 Fn 𝑋) |
6 | 1, 4, 5 | 3syl 18 |
. 2
⊢ (𝜑 → 𝐾 Fn 𝑋) |
7 | | frgpup.n |
. . . 4
⊢ 𝑁 = (invg‘𝐻) |
8 | | frgpup.t |
. . . 4
⊢ 𝑇 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦)))) |
9 | | frgpup.h |
. . . 4
⊢ (𝜑 → 𝐻 ∈ Grp) |
10 | | frgpup.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
11 | | frgpup.a |
. . . 4
⊢ (𝜑 → 𝐹:𝐼⟶𝐵) |
12 | | frgpup.w |
. . . 4
⊢ 𝑊 = ( I ‘Word (𝐼 ×
2o)) |
13 | | frgpup.r |
. . . 4
⊢ ∼ = (
~FG ‘𝐼) |
14 | | frgpup.g |
. . . 4
⊢ 𝐺 = (freeGrp‘𝐼) |
15 | | frgpup.e |
. . . 4
⊢ 𝐸 = ran (𝑔 ∈ 𝑊 ↦ 〈[𝑔] ∼ , (𝐻 Σg
(𝑇 ∘ 𝑔))〉) |
16 | 3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15 | frgpup1 19369 |
. . 3
⊢ (𝜑 → 𝐸 ∈ (𝐺 GrpHom 𝐻)) |
17 | 2, 3 | ghmf 18826 |
. . 3
⊢ (𝐸 ∈ (𝐺 GrpHom 𝐻) → 𝐸:𝑋⟶𝐵) |
18 | | ffn 6593 |
. . 3
⊢ (𝐸:𝑋⟶𝐵 → 𝐸 Fn 𝑋) |
19 | 16, 17, 18 | 3syl 18 |
. 2
⊢ (𝜑 → 𝐸 Fn 𝑋) |
20 | | eqid 2738 |
. . . . . . . . 9
⊢
(freeMnd‘(𝐼
× 2o)) = (freeMnd‘(𝐼 × 2o)) |
21 | 14, 20, 13 | frgpval 19352 |
. . . . . . . 8
⊢ (𝐼 ∈ 𝑉 → 𝐺 = ((freeMnd‘(𝐼 × 2o))
/s ∼ )) |
22 | 10, 21 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐺 = ((freeMnd‘(𝐼 × 2o))
/s ∼ )) |
23 | | 2on 8299 |
. . . . . . . . . . 11
⊢
2o ∈ On |
24 | | xpexg 7591 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ 𝑉 ∧ 2o ∈ On) →
(𝐼 × 2o)
∈ V) |
25 | 10, 23, 24 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐼 × 2o) ∈
V) |
26 | | wrdexg 14215 |
. . . . . . . . . 10
⊢ ((𝐼 × 2o) ∈ V
→ Word (𝐼 ×
2o) ∈ V) |
27 | | fvi 6837 |
. . . . . . . . . 10
⊢ (Word
(𝐼 × 2o)
∈ V → ( I ‘Word (𝐼 × 2o)) = Word (𝐼 ×
2o)) |
28 | 25, 26, 27 | 3syl 18 |
. . . . . . . . 9
⊢ (𝜑 → ( I ‘Word (𝐼 × 2o)) = Word
(𝐼 ×
2o)) |
29 | 12, 28 | eqtrid 2790 |
. . . . . . . 8
⊢ (𝜑 → 𝑊 = Word (𝐼 × 2o)) |
30 | | eqid 2738 |
. . . . . . . . . 10
⊢
(Base‘(freeMnd‘(𝐼 × 2o))) =
(Base‘(freeMnd‘(𝐼 × 2o))) |
31 | 20, 30 | frmdbas 18479 |
. . . . . . . . 9
⊢ ((𝐼 × 2o) ∈ V
→ (Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 ×
2o)) |
32 | 25, 31 | syl 17 |
. . . . . . . 8
⊢ (𝜑 →
(Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 ×
2o)) |
33 | 29, 32 | eqtr4d 2781 |
. . . . . . 7
⊢ (𝜑 → 𝑊 = (Base‘(freeMnd‘(𝐼 ×
2o)))) |
34 | 13 | fvexi 6781 |
. . . . . . . 8
⊢ ∼ ∈
V |
35 | 34 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → ∼ ∈
V) |
36 | | fvexd 6782 |
. . . . . . 7
⊢ (𝜑 → (freeMnd‘(𝐼 × 2o)) ∈
V) |
37 | 22, 33, 35, 36 | qusbas 17244 |
. . . . . 6
⊢ (𝜑 → (𝑊 / ∼ ) =
(Base‘𝐺)) |
38 | 2, 37 | eqtr4id 2797 |
. . . . 5
⊢ (𝜑 → 𝑋 = (𝑊 / ∼ )) |
39 | | eqimss 3977 |
. . . . 5
⊢ (𝑋 = (𝑊 / ∼ ) → 𝑋 ⊆ (𝑊 / ∼ )) |
40 | 38, 39 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑋 ⊆ (𝑊 / ∼ )) |
41 | 40 | sselda 3921 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → 𝑎 ∈ (𝑊 / ∼ )) |
42 | | eqid 2738 |
. . . 4
⊢ (𝑊 / ∼ ) = (𝑊 / ∼ ) |
43 | | fveq2 6767 |
. . . . 5
⊢ ([𝑡] ∼ = 𝑎 → (𝐾‘[𝑡] ∼ ) = (𝐾‘𝑎)) |
44 | | fveq2 6767 |
. . . . 5
⊢ ([𝑡] ∼ = 𝑎 → (𝐸‘[𝑡] ∼ ) = (𝐸‘𝑎)) |
45 | 43, 44 | eqeq12d 2754 |
. . . 4
⊢ ([𝑡] ∼ = 𝑎 → ((𝐾‘[𝑡] ∼ ) = (𝐸‘[𝑡] ∼ ) ↔ (𝐾‘𝑎) = (𝐸‘𝑎))) |
46 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 ∈ 𝑊) |
47 | 29 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 = Word (𝐼 × 2o)) |
48 | 46, 47 | eleqtrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 ∈ Word (𝐼 × 2o)) |
49 | | wrdf 14210 |
. . . . . . . . . . . . 13
⊢ (𝑡 ∈ Word (𝐼 × 2o) → 𝑡:(0..^(♯‘𝑡))⟶(𝐼 × 2o)) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡:(0..^(♯‘𝑡))⟶(𝐼 × 2o)) |
51 | 50 | ffvelrnda 6954 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → (𝑡‘𝑛) ∈ (𝐼 × 2o)) |
52 | | elxp2 5609 |
. . . . . . . . . . 11
⊢ ((𝑡‘𝑛) ∈ (𝐼 × 2o) ↔ ∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2o (𝑡‘𝑛) = 〈𝑖, 𝑗〉) |
53 | 51, 52 | sylib 217 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → ∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2o (𝑡‘𝑛) = 〈𝑖, 𝑗〉) |
54 | | fveq2 6767 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑖 → (𝐹‘𝑦) = (𝐹‘𝑖)) |
55 | 54 | fveq2d 6771 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = 𝑖 → (𝑁‘(𝐹‘𝑦)) = (𝑁‘(𝐹‘𝑖))) |
56 | 54, 55 | ifeq12d 4481 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑖 → if(𝑧 = ∅, (𝐹‘𝑦), (𝑁‘(𝐹‘𝑦))) = if(𝑧 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖)))) |
57 | | eqeq1 2742 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 = 𝑗 → (𝑧 = ∅ ↔ 𝑗 = ∅)) |
58 | 57 | ifbid 4483 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = 𝑗 → if(𝑧 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖)))) |
59 | | fvex 6780 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹‘𝑖) ∈ V |
60 | | fvex 6780 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁‘(𝐹‘𝑖)) ∈ V |
61 | 59, 60 | ifex 4510 |
. . . . . . . . . . . . . . . 16
⊢ if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) ∈ V |
62 | 56, 58, 8, 61 | ovmpo 7424 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2o) → (𝑖𝑇𝑗) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖)))) |
63 | 62 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2o)) → (𝑖𝑇𝑗) = if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖)))) |
64 | | elpri 4584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ {∅, 1o}
→ (𝑗 = ∅ ∨
𝑗 =
1o)) |
65 | | df2o3 8293 |
. . . . . . . . . . . . . . . . 17
⊢
2o = {∅, 1o} |
66 | 64, 65 | eleq2s 2857 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ 2o →
(𝑗 = ∅ ∨ 𝑗 =
1o)) |
67 | | frgpup3.e |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐾 ∘ 𝑈) = 𝐹) |
68 | 67 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐾 ∘ 𝑈) = 𝐹) |
69 | 68 | fveq1d 6769 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐹‘𝑖)) |
70 | | frgpup.u |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 𝑈 =
(varFGrp‘𝐼) |
71 | 13, 70, 14, 2 | vrgpf 19362 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐼 ∈ 𝑉 → 𝑈:𝐼⟶𝑋) |
72 | 10, 71 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑈:𝐼⟶𝑋) |
73 | | fvco3 6860 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑈:𝐼⟶𝑋 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐾‘(𝑈‘𝑖))) |
74 | 72, 73 | sylan 580 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((𝐾 ∘ 𝑈)‘𝑖) = (𝐾‘(𝑈‘𝑖))) |
75 | 69, 74 | eqtr3d 2780 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐹‘𝑖) = (𝐾‘(𝑈‘𝑖))) |
76 | 75 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝐹‘𝑖) = (𝐾‘(𝑈‘𝑖))) |
77 | | iftrue 4466 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = ∅ → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐹‘𝑖)) |
78 | 77 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐹‘𝑖)) |
79 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → 𝑗 = ∅) |
80 | 79 | opeq2d 4812 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → 〈𝑖, 𝑗〉 = 〈𝑖, ∅〉) |
81 | 80 | s1eqd 14294 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → 〈“〈𝑖, 𝑗〉”〉 =
〈“〈𝑖,
∅〉”〉) |
82 | 81 | eceq1d 8525 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → [〈“〈𝑖, 𝑗〉”〉] ∼ =
[〈“〈𝑖,
∅〉”〉] ∼ ) |
83 | 13, 70 | vrgpval 19361 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) = [〈“〈𝑖, ∅〉”〉] ∼
) |
84 | 10, 83 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) = [〈“〈𝑖, ∅〉”〉] ∼
) |
85 | 84 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝑈‘𝑖) = [〈“〈𝑖, ∅〉”〉] ∼
) |
86 | 82, 85 | eqtr4d 2781 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → [〈“〈𝑖, 𝑗〉”〉] ∼ = (𝑈‘𝑖)) |
87 | 86 | fveq2d 6771 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ ) = (𝐾‘(𝑈‘𝑖))) |
88 | 76, 78, 87 | 3eqtr4d 2788 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = ∅) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
89 | 75 | fveq2d 6771 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑁‘(𝐹‘𝑖)) = (𝑁‘(𝐾‘(𝑈‘𝑖)))) |
90 | 72 | ffvelrnda 6954 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑈‘𝑖) ∈ 𝑋) |
91 | | eqid 2738 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(invg‘𝐺) = (invg‘𝐺) |
92 | 2, 91, 7 | ghminv 18829 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ (𝐺 GrpHom 𝐻) ∧ (𝑈‘𝑖) ∈ 𝑋) → (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))) = (𝑁‘(𝐾‘(𝑈‘𝑖)))) |
93 | 1, 90, 92 | syl2an2r 682 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖))) = (𝑁‘(𝐾‘(𝑈‘𝑖)))) |
94 | 89, 93 | eqtr4d 2781 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → (𝑁‘(𝐹‘𝑖)) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖)))) |
95 | 94 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → (𝑁‘(𝐹‘𝑖)) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖)))) |
96 | | 1n0 8306 |
. . . . . . . . . . . . . . . . . . . 20
⊢
1o ≠ ∅ |
97 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → 𝑗 = 1o) |
98 | 97 | neeq1d 3003 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → (𝑗 ≠ ∅ ↔ 1o ≠
∅)) |
99 | 96, 98 | mpbiri 257 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → 𝑗 ≠ ∅) |
100 | | ifnefalse 4472 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ≠ ∅ → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝑁‘(𝐹‘𝑖))) |
101 | 99, 100 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝑁‘(𝐹‘𝑖))) |
102 | 97 | opeq2d 4812 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → 〈𝑖, 𝑗〉 = 〈𝑖, 1o〉) |
103 | 102 | s1eqd 14294 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) →
〈“〈𝑖, 𝑗〉”〉 =
〈“〈𝑖,
1o〉”〉) |
104 | 103 | eceq1d 8525 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) →
[〈“〈𝑖,
𝑗〉”〉] ∼ =
[〈“〈𝑖,
1o〉”〉] ∼ ) |
105 | 13, 70, 14, 91 | vrgpinv 19363 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐼 ∈ 𝑉 ∧ 𝑖 ∈ 𝐼) → ((invg‘𝐺)‘(𝑈‘𝑖)) = [〈“〈𝑖, 1o〉”〉] ∼
) |
106 | 10, 105 | sylan 580 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑖 ∈ 𝐼) → ((invg‘𝐺)‘(𝑈‘𝑖)) = [〈“〈𝑖, 1o〉”〉] ∼
) |
107 | 106 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) →
((invg‘𝐺)‘(𝑈‘𝑖)) = [〈“〈𝑖, 1o〉”〉] ∼
) |
108 | 104, 107 | eqtr4d 2781 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) →
[〈“〈𝑖,
𝑗〉”〉] ∼ =
((invg‘𝐺)‘(𝑈‘𝑖))) |
109 | 108 | fveq2d 6771 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ ) = (𝐾‘((invg‘𝐺)‘(𝑈‘𝑖)))) |
110 | 95, 101, 109 | 3eqtr4d 2788 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 = 1o) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
111 | 88, 110 | jaodan 955 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ (𝑗 = ∅ ∨ 𝑗 = 1o)) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
112 | 66, 111 | sylan2 593 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑖 ∈ 𝐼) ∧ 𝑗 ∈ 2o) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
113 | 112 | anasss 467 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2o)) → if(𝑗 = ∅, (𝐹‘𝑖), (𝑁‘(𝐹‘𝑖))) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
114 | 63, 113 | eqtrd 2778 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2o)) → (𝑖𝑇𝑗) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
115 | | fveq2 6767 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝑇‘(𝑡‘𝑛)) = (𝑇‘〈𝑖, 𝑗〉)) |
116 | | df-ov 7271 |
. . . . . . . . . . . . . . 15
⊢ (𝑖𝑇𝑗) = (𝑇‘〈𝑖, 𝑗〉) |
117 | 115, 116 | eqtr4di 2796 |
. . . . . . . . . . . . . 14
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝑇‘(𝑡‘𝑛)) = (𝑖𝑇𝑗)) |
118 | | s1eq 14293 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → 〈“(𝑡‘𝑛)”〉 = 〈“〈𝑖, 𝑗〉”〉) |
119 | 118 | eceq1d 8525 |
. . . . . . . . . . . . . . 15
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → [〈“(𝑡‘𝑛)”〉] ∼ =
[〈“〈𝑖,
𝑗〉”〉] ∼
) |
120 | 119 | fveq2d 6771 |
. . . . . . . . . . . . . 14
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼ ) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼ )) |
121 | 117, 120 | eqeq12d 2754 |
. . . . . . . . . . . . 13
⊢ ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → ((𝑇‘(𝑡‘𝑛)) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼ ) ↔ (𝑖𝑇𝑗) = (𝐾‘[〈“〈𝑖, 𝑗〉”〉] ∼
))) |
122 | 114, 121 | syl5ibrcom 246 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑖 ∈ 𝐼 ∧ 𝑗 ∈ 2o)) → ((𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼
))) |
123 | 122 | rexlimdvva 3221 |
. . . . . . . . . . 11
⊢ (𝜑 → (∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2o (𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼
))) |
124 | 123 | ad2antrr 723 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → (∃𝑖 ∈ 𝐼 ∃𝑗 ∈ 2o (𝑡‘𝑛) = 〈𝑖, 𝑗〉 → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼
))) |
125 | 53, 124 | mpd 15 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → (𝑇‘(𝑡‘𝑛)) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼ )) |
126 | 125 | mpteq2dva 5174 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛))) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼
))) |
127 | 3, 7, 8, 9, 10, 11 | frgpuptf 19364 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇:(𝐼 × 2o)⟶𝐵) |
128 | | fcompt 6998 |
. . . . . . . . 9
⊢ ((𝑇:(𝐼 × 2o)⟶𝐵 ∧ 𝑡:(0..^(♯‘𝑡))⟶(𝐼 × 2o)) → (𝑇 ∘ 𝑡) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛)))) |
129 | 127, 50, 128 | syl2an2r 682 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝑇‘(𝑡‘𝑛)))) |
130 | 51 | s1cld 14296 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → 〈“(𝑡‘𝑛)”〉 ∈ Word (𝐼 × 2o)) |
131 | 29 | ad2antrr 723 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → 𝑊 = Word (𝐼 × 2o)) |
132 | 130, 131 | eleqtrrd 2842 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → 〈“(𝑡‘𝑛)”〉 ∈ 𝑊) |
133 | 14, 13, 12, 2 | frgpeccl 19355 |
. . . . . . . . . 10
⊢
(〈“(𝑡‘𝑛)”〉 ∈ 𝑊 → [〈“(𝑡‘𝑛)”〉] ∼ ∈ 𝑋) |
134 | 132, 133 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑡 ∈ 𝑊) ∧ 𝑛 ∈ (0..^(♯‘𝑡))) → [〈“(𝑡‘𝑛)”〉] ∼ ∈ 𝑋) |
135 | 50 | feqmptd 6830 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑡 = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝑡‘𝑛))) |
136 | 10 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐼 ∈ 𝑉) |
137 | 136, 23, 24 | sylancl 586 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐼 × 2o) ∈
V) |
138 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢
(varFMnd‘(𝐼 × 2o)) =
(varFMnd‘(𝐼 × 2o)) |
139 | 138 | vrmdfval 18483 |
. . . . . . . . . . . 12
⊢ ((𝐼 × 2o) ∈ V
→ (varFMnd‘(𝐼 × 2o)) = (𝑤 ∈ (𝐼 × 2o) ↦
〈“𝑤”〉)) |
140 | 137, 139 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
(varFMnd‘(𝐼 × 2o)) = (𝑤 ∈ (𝐼 × 2o) ↦
〈“𝑤”〉)) |
141 | | s1eq 14293 |
. . . . . . . . . . 11
⊢ (𝑤 = (𝑡‘𝑛) → 〈“𝑤”〉 = 〈“(𝑡‘𝑛)”〉) |
142 | 51, 135, 140, 141 | fmptco 6994 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ 〈“(𝑡‘𝑛)”〉)) |
143 | | eqidd 2739 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )) |
144 | | eceq1 8524 |
. . . . . . . . . 10
⊢ (𝑤 = 〈“(𝑡‘𝑛)”〉 → [𝑤] ∼ =
[〈“(𝑡‘𝑛)”〉] ∼ ) |
145 | 132, 142,
143, 144 | fmptco 6994 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ [〈“(𝑡‘𝑛)”〉] ∼ )) |
146 | 1 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 ∈ (𝐺 GrpHom 𝐻)) |
147 | 146, 4 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾:𝑋⟶𝐵) |
148 | 147 | feqmptd 6830 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 = (𝑤 ∈ 𝑋 ↦ (𝐾‘𝑤))) |
149 | | fveq2 6767 |
. . . . . . . . 9
⊢ (𝑤 = [〈“(𝑡‘𝑛)”〉] ∼ → (𝐾‘𝑤) = (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼ )) |
150 | 134, 145,
148, 149 | fmptco 6994 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))) = (𝑛 ∈ (0..^(♯‘𝑡)) ↦ (𝐾‘[〈“(𝑡‘𝑛)”〉] ∼
))) |
151 | 126, 129,
150 | 3eqtr4d 2788 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑇 ∘ 𝑡) = (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) |
152 | 151 | oveq2d 7284 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐻 Σg (𝑇 ∘ 𝑡)) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))))) |
153 | 3, 7, 8, 9, 10, 11, 12, 13, 14, 2, 15 | frgpupval 19368 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐻 Σg
(𝑇 ∘ 𝑡))) |
154 | | ghmmhm 18832 |
. . . . . . . 8
⊢ (𝐾 ∈ (𝐺 GrpHom 𝐻) → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
155 | 146, 154 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝐾 ∈ (𝐺 MndHom 𝐻)) |
156 | 138 | vrmdf 18485 |
. . . . . . . . . . 11
⊢ ((𝐼 × 2o) ∈ V
→ (varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶Word (𝐼 ×
2o)) |
157 | 137, 156 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
(varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶Word (𝐼 ×
2o)) |
158 | 47 | feq3d 6580 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
((varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶𝑊 ↔
(varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶Word (𝐼 ×
2o))) |
159 | 157, 158 | mpbird 256 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
(varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶𝑊) |
160 | | wrdco 14532 |
. . . . . . . . 9
⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧
(varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶𝑊) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word 𝑊) |
161 | 48, 159, 160 | syl2anc 584 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word 𝑊) |
162 | 33 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 = (Base‘(freeMnd‘(𝐼 ×
2o)))) |
163 | 162 | mpteq1d 5169 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈
(Base‘(freeMnd‘(𝐼 × 2o))) ↦ [𝑤] ∼ )) |
164 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢ (𝑤 ∈
(Base‘(freeMnd‘(𝐼 × 2o))) ↦ [𝑤] ∼ ) = (𝑤 ∈
(Base‘(freeMnd‘(𝐼 × 2o))) ↦ [𝑤] ∼ ) |
165 | 20, 30, 14, 13, 164 | frgpmhm 19359 |
. . . . . . . . . . . 12
⊢ (𝐼 ∈ 𝑉 → (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))
↦ [𝑤] ∼ )
∈ ((freeMnd‘(𝐼
× 2o)) MndHom 𝐺)) |
166 | 136, 165 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ (Base‘(freeMnd‘(𝐼 × 2o)))
↦ [𝑤] ∼ )
∈ ((freeMnd‘(𝐼
× 2o)) MndHom 𝐺)) |
167 | 163, 166 | eqeltrd 2839 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈
((freeMnd‘(𝐼 ×
2o)) MndHom 𝐺)) |
168 | 30, 2 | mhmf 18423 |
. . . . . . . . . 10
⊢ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈
((freeMnd‘(𝐼 ×
2o)) MndHom 𝐺)
→ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼
):(Base‘(freeMnd‘(𝐼 × 2o)))⟶𝑋) |
169 | 167, 168 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼
):(Base‘(freeMnd‘(𝐼 × 2o)))⟶𝑋) |
170 | 162 | feq2d 6579 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋 ↔ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼
):(Base‘(freeMnd‘(𝐼 × 2o)))⟶𝑋)) |
171 | 169, 170 | mpbird 256 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋) |
172 | | wrdco 14532 |
. . . . . . . 8
⊢
((((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word 𝑊 ∧ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ):𝑊⟶𝑋) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) ∈ Word 𝑋) |
173 | 161, 171,
172 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) ∈ Word 𝑋) |
174 | 2 | gsumwmhm 18472 |
. . . . . . 7
⊢ ((𝐾 ∈ (𝐺 MndHom 𝐻) ∧ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) ∈ Word 𝑋) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))))) |
175 | 155, 173,
174 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) = (𝐻 Σg (𝐾 ∘ ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))))) |
176 | 152, 153,
175 | 3eqtr4d 2788 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐸‘[𝑡] ∼ ) = (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))))) |
177 | 20, 138 | frmdgsum 18489 |
. . . . . . . . 9
⊢ (((𝐼 × 2o) ∈ V
∧ 𝑡 ∈ Word (𝐼 × 2o)) →
((freeMnd‘(𝐼 ×
2o)) Σg
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) = 𝑡) |
178 | 25, 48, 177 | syl2an2r 682 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((freeMnd‘(𝐼 × 2o))
Σg ((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)) = 𝑡) |
179 | 178 | fveq2d 6771 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼
)‘((freeMnd‘(𝐼
× 2o)) Σg
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))) = ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘𝑡)) |
180 | | wrdco 14532 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ Word (𝐼 × 2o) ∧
(varFMnd‘(𝐼 × 2o)):(𝐼 × 2o)⟶Word (𝐼 × 2o)) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word Word (𝐼 ×
2o)) |
181 | 48, 157, 180 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word Word (𝐼 ×
2o)) |
182 | 32 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (Base‘(freeMnd‘(𝐼 × 2o))) = Word
(𝐼 ×
2o)) |
183 | | wrdeq 14227 |
. . . . . . . . . 10
⊢
((Base‘(freeMnd‘(𝐼 × 2o))) = Word (𝐼 × 2o) →
Word (Base‘(freeMnd‘(𝐼 × 2o))) = Word Word (𝐼 ×
2o)) |
184 | 182, 183 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → Word
(Base‘(freeMnd‘(𝐼 × 2o))) = Word Word (𝐼 ×
2o)) |
185 | 181, 184 | eleqtrrd 2842 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) →
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word
(Base‘(freeMnd‘(𝐼 × 2o)))) |
186 | 30 | gsumwmhm 18472 |
. . . . . . . 8
⊢ (((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∈
((freeMnd‘(𝐼 ×
2o)) MndHom 𝐺)
∧ ((varFMnd‘(𝐼 × 2o)) ∘ 𝑡) ∈ Word
(Base‘(freeMnd‘(𝐼 × 2o)))) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼
)‘((freeMnd‘(𝐼
× 2o)) Σg
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))) = (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) |
187 | 167, 185,
186 | syl2anc 584 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼
)‘((freeMnd‘(𝐼
× 2o)) Σg
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))) = (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) |
188 | 12, 13 | efger 19312 |
. . . . . . . . 9
⊢ ∼ Er
𝑊 |
189 | 188 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ∼ Er 𝑊) |
190 | 12 | fvexi 6781 |
. . . . . . . . 9
⊢ 𝑊 ∈ V |
191 | 190 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → 𝑊 ∈ V) |
192 | | eqid 2738 |
. . . . . . . 8
⊢ (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) = (𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) |
193 | 189, 191,
192 | divsfval 17246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ )‘𝑡) = [𝑡] ∼ ) |
194 | 179, 187,
193 | 3eqtr3d 2786 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡))) = [𝑡] ∼ ) |
195 | 194 | fveq2d 6771 |
. . . . 5
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘(𝐺 Σg ((𝑤 ∈ 𝑊 ↦ [𝑤] ∼ ) ∘
((varFMnd‘(𝐼 × 2o)) ∘ 𝑡)))) = (𝐾‘[𝑡] ∼ )) |
196 | 176, 195 | eqtr2d 2779 |
. . . 4
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑊) → (𝐾‘[𝑡] ∼ ) = (𝐸‘[𝑡] ∼ )) |
197 | 42, 45, 196 | ectocld 8561 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ (𝑊 / ∼ )) → (𝐾‘𝑎) = (𝐸‘𝑎)) |
198 | 41, 197 | syldan 591 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑋) → (𝐾‘𝑎) = (𝐸‘𝑎)) |
199 | 6, 19, 198 | eqfnfvd 6905 |
1
⊢ (𝜑 → 𝐾 = 𝐸) |